# Properties

 Label 2-15e2-1.1-c1-0-3 Degree $2$ Conductor $225$ Sign $-1$ Analytic cond. $1.79663$ Root an. cond. $1.34038$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·2-s + 2·4-s − 3·7-s − 2·11-s + 13-s + 6·14-s − 4·16-s − 2·17-s − 5·19-s + 4·22-s − 6·23-s − 2·26-s − 6·28-s − 10·29-s − 3·31-s + 8·32-s + 4·34-s + 2·37-s + 10·38-s + 8·41-s + 43-s − 4·44-s + 12·46-s − 2·47-s + 2·49-s + 2·52-s + 4·53-s + ⋯
 L(s)  = 1 − 1.41·2-s + 4-s − 1.13·7-s − 0.603·11-s + 0.277·13-s + 1.60·14-s − 16-s − 0.485·17-s − 1.14·19-s + 0.852·22-s − 1.25·23-s − 0.392·26-s − 1.13·28-s − 1.85·29-s − 0.538·31-s + 1.41·32-s + 0.685·34-s + 0.328·37-s + 1.62·38-s + 1.24·41-s + 0.152·43-s − 0.603·44-s + 1.76·46-s − 0.291·47-s + 2/7·49-s + 0.277·52-s + 0.549·53-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $-1$ Analytic conductor: $$1.79663$$ Root analytic conductor: $$1.34038$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 225,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 + p T + p T^{2}$$
7 $$1 + 3 T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 - T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 + 5 T + p T^{2}$$
23 $$1 + 6 T + p T^{2}$$
29 $$1 + 10 T + p T^{2}$$
31 $$1 + 3 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 - 8 T + p T^{2}$$
43 $$1 - T + p T^{2}$$
47 $$1 + 2 T + p T^{2}$$
53 $$1 - 4 T + p T^{2}$$
59 $$1 - 10 T + p T^{2}$$
61 $$1 - 7 T + p T^{2}$$
67 $$1 + 3 T + p T^{2}$$
71 $$1 - 8 T + p T^{2}$$
73 $$1 + 14 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + 6 T + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 - 17 T + p T^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−11.36506854345797939837793722937, −10.51847852466115917559094220478, −9.725369907410972008022337048795, −8.938558667143995354006242020404, −7.965886472126962800720761127653, −6.96141726621690601574492077524, −5.87930303915993667052597404250, −4.01465426298836833468759021488, −2.23186561512245260821665501913, 0, 2.23186561512245260821665501913, 4.01465426298836833468759021488, 5.87930303915993667052597404250, 6.96141726621690601574492077524, 7.965886472126962800720761127653, 8.938558667143995354006242020404, 9.725369907410972008022337048795, 10.51847852466115917559094220478, 11.36506854345797939837793722937